# Relative $K_0$ and relative cycle class map

## TL;DR

This paper investigates the relative $K_0$ groups of exact and triangulated categories and introduces a cycle class map connecting Chow groups with modulus to relative $K_0$, advancing understanding in algebraic K-theory.

## Contribution

It constructs a new cycle class map from Chow groups with modulus to relative $K_0$, linking algebraic cycles and K-theory in a relative setting.

## Key findings

- Established properties of relative $K_0$ in exact and triangulated categories
- Constructed a cycle class map from Chow groups with modulus to relative $K_0$
- Provided applications to algebraic K-theory and cycle theory

## Abstract

We study relative $K_0$ of exact categories and triangulated categories. As an application, we construct a cycle class map from Chow groups with modulus to relative $K_0$.